Cardinality of a finite set

Definition

Let $A$ be a set and let $n$ be a natural number. The set $A$ has $n$ elements if there is a bijection from $\{1, 2, \dots, n\}$ to $A$. For $n = 0$, the set $\{1, \dots, n\}$ is the empty set, so only the empty set has $0$ elements. A set is finite if it has $n$ elements for some natural number $n$.

For a finite set $A$, this number $n$ is unique. It is the cardinality of $A$, with the notation $|A|$ or $\# A$.